Author |
Topic: The Number Pyramid (Read 6909 times) |
|
Naumz
Newbie
Rule over your dreams. Don't be ruled by them.
 
Gender: 
Posts: 17
|
 |
The Number Pyramid
« on: Oct 3rd, 2006, 6:15pm » |
 Quote  Modify
|
A Number Pyramid is composed of the 10 different numbers 0-9 with a top row of 1 number resting on a second row of 2 sitting on a third row of 3 supported by a bottom row of 4. For example, a Number Pyramid could be:
0
1 2
3 4 5
6 7 8 9
Given the clues below, can you determine the composition of the Number Pyramid?
a) The number at the top of Number Pyramid minus the leftmost number in row 2 equals 5.
b) Both the four leftmost numbers in each row and the three numbers in row 3 sum to 14.
c) In the bottom row of the pyramid, the second number from the left minus the leftmost number equals 5.
d) The rightmost number in row 3 minus the rightmost number in row 4 equals 3.
e) The four numbers in the bottom row sum to 17.
|
| « Last Edit: Oct 4th, 2006, 6:15pm by Icarus » |
 IP Logged |
|
|
|
Naumz
Newbie
Rule over your dreams. Don't be ruled by them.
Gender:
Posts: 17
|
 |
Re: The Number Pyramid
« Reply #1 on: Oct 3rd, 2006, 6:16pm » |
Quote Modify
|
The last row is shifted a bit to the right, not that it matters, of course. But the formatting looks weird...loll 
|
|
IP Logged |
|
|
|
towr
wu::riddles Moderator
Uberpuzzler
Some people are average, some are just mean.
Gender:
Posts: 13730
|
|
Re: The Number Pyramid
« Reply #2 on: Oct 4th, 2006, 12:19am » |
Quote Modify
|
Seems a simple matter of solving a system of equations.
with a further constraint all numbers 0-9 are used.
a-b=5
a+b+d+g=14 & d+e+f=14
h-g=5
f-j=3
g+h+i+j=17
A computer would get it in a second, of course..
|
|
IP Logged |
Wikipedia, Google, Mathworld, Integer sequence DB
|
|
|
Naumz
Newbie
Rule over your dreams. Don't be ruled by them.
Gender:
Posts: 17
|
|
Re: The Number Pyramid
« Reply #3 on: Oct 4th, 2006, 11:22am » |
Quote Modify
|
I did solve it by a computer. But i thought there would be a smipler way to do by ourselves without solving the equations completely...guess i was wrong. I actually got this as part of a project.
|
|
IP Logged |
|
|
|
towr
wu::riddles Moderator
Uberpuzzler
Some people are average, some are just mean.
Gender:
Posts: 13730
|
|
Re: The Number Pyramid
« Reply #4 on: Oct 4th, 2006, 12:59pm » |
Quote Modify
|
It's probably not impossible to do it by hand. It's just that I'm lazy, and would rather use a simple solver than think twice 
The standard way to do it by hand would probably to make a 10 by 10 table (variables vertically, digits horizontally) and start eliminating ranges that don't work.
e.g. a-b = 5 gives 5 =< a =< 9 and 0 =< b =< 4
It's tricky for multiple variables in an equation though.
|
|
IP Logged |
Wikipedia, Google, Mathworld, Integer sequence DB
|
|
|
Icarus
wu::riddles Moderator
Uberpuzzler
Boldly going where even angels fear to tread.
Gender:
Posts: 4863
|
|
Re: The Number Pyramid
« Reply #5 on: Oct 4th, 2006, 6:18pm » |
Quote Modify
|
on Oct 3rd, 2006, 6:16pm, Naumz wrote:| The last row is shifted a bit to the right, not that it matters, of course. But the formatting looks weird...loll |
|
I reformatted it for you. YaBB does a lot of weird things to formatted text. In particular, it doesn't allow more than about 5 or 6 spaces together. When it sees that many it always replaces them with a single space. Thats why I removed the "indent". (I also used a constant width font, so that spaces are just as wide as the characters.)
|
|
IP Logged |
"Pi goes on and on and on ...
And e is just as cursed.
I wonder: Which is larger
When their digits are reversed? " - Anonymous
|
|
|
Icarus
wu::riddles Moderator
Uberpuzzler
Boldly going where even angels fear to tread.
Gender:
Posts: 4863
|
|
Re: The Number Pyramid
« Reply #6 on: Oct 4th, 2006, 6:32pm » |
Quote Modify
|
Since you only have 6 equations, the best you can do is solve for 6 variables in terms of the other 4. Since one variable (c) is not involved in any of the equations, we have 3 variables that determine the other 6, and one left to take on whatever value is left over.
While the equations place some limitations on what values the 3 "independent" variables take on (for instance, g has to be >= 3), for the most part, at this stage, you are reduced to using brute force to finish solving the puzzle by simply trying out triplets of values and see which ones result in all values from 0-9 except one in the 3 independent and 6 dependent variables (the remaining value is the value of c).
And since brute force is what is called for at this stage, what better way to apply it than to use a computer?
|
|
IP Logged |
"Pi goes on and on and on ...
And e is just as cursed.
I wonder: Which is larger
When their digits are reversed? " - Anonymous
|
|
|
SWF
Uberpuzzler
Posts: 879
|
|
Re: The Number Pyramid
« Reply #7 on: Oct 4th, 2006, 6:48pm » |
Quote Modify
|
In addition to the equations towr gave, there is also 45=a+b+c+d+e+f+g+h+i+j. From this subtract some of towr's equations to get c=d+g. Substitute to remove h, a, d, and g from the equations:
(1) 2*b+c=9
(2) c-g+e+j=11
(3) 2*g+i+j=12
From (1) and a-b=5, b can only be 0, 1, 2 and the corresponding values of c are 9, 7, or 5 (b can't be 3 or c would be 3 too, and b can't be 4 because c would be 1 and d or g would also have to be 1). Since all letters must have different values, c=d+g, g<5 (because h=g+5) there are not many possibilities (keeping in mind that h=g+5 must also have a different value):
b=0, c=9, (d=8,g=1 or d=6,g=3)
b=1, c=7, (d=4,g=3 or d=3,g=4)
b=2, c=5, (d=1,g=4 or d=4,g=1)
Eliminate j from (2) and (3) to get: 1+c-3*g=i-e. Plug in the possible values of c and g into this equation and try to find acceptable values of i and e that do not duplicate any of the other numbers (make sure i and e are also different from h=g+5 and a=b+5). The remaining possibilites are now:
b=0, c=9, d=6, g=3, i=2, e=1
b=2, c=5, d=1, g=4, i=0, e=6
b=2, c=5, d=4, g=1, i=3, e=0
Only the row with b=0 permits values of j and f with f=j+3 which do not duplicate other numbers (j=4, f=7), and the only acceptable values are
a=5, b=0, c=9, d=6, e=1, f=7, g=3, h=8, i=2, j=4
Hmmm, that seemed much easier before I tried to type out a full explanation.
|
|
IP Logged |
|
|
|
|
RIDDLES SITE
WRITE MATH!
Home 
Help 
Search 
Members 
Login 
Register
Reply
Notify of replies
Send Topic
Print